Derivative Rules
The Derivative tells us the slope of a function at any point.
There are rules we can follow to find many derivatives.
For example:
- The slope of a constant value (like 3) is always 0
- The slope of a line like 2x is 2, or 3x is 3 etc
- and so on.
Here are useful rules to help you work out the derivatives of many functions (with examples below). Note: the little mark ’ means derivative of, and f and g are functions.
Common Functions | Function |
Derivative |
---|---|---|
Constant | c | 0 |
Line | x | 1 |
ax | a | |
Square | x^{2} | 2x |
Square Root | √x | (½)x^{-½} |
Exponential | e^{x} | e^{x} |
a^{x} | ln(a) a^{x} | |
Logarithms | ln(x) | 1/x |
log_{a}(x) | 1 / (x ln(a)) | |
Trigonometry (x is in radians) | sin(x) | cos(x) |
cos(x) | −sin(x) | |
tan(x) | sec^{2}(x) | |
Inverse Trigonometry | sin^{-1}(x) | 1/√(1−x^{2}) |
cos^{-1}(x) | −1/√(1−x^{2}) | |
tan^{-1}(x) | 1/(1+x^{2}) | |
Rules | Function |
Derivative |
Multiplication by constant | cf | cf’ |
Power Rule | x^{n} | nx^{n−1} |
Sum Rule | f + g | f’ + g’ |
Difference Rule | f - g | f’ − g’ |
Product Rule | fg | f g’ + f’ g |
Quotient Rule | f/g | \frac{f’ g − g’ f}{g^{2}} |
Reciprocal Rule | 1/f | −f’/f^{2} |
Chain Rule (as "Composition of Functions") |
f º g | (f’ º g) × g’ |
Chain Rule (using ’ ) | f(g(x)) | f’(g(x))g’(x) |
Chain Rule (using \frac{d}{dx} ) | \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx} |
"The derivative of" is also written \frac{d}{dx}
So \frac{d}{dx}sin(x) and sin(x)’ both mean "The derivative of sin(x)"
Examples
Example: what is the derivative of sin(x) ?
From the table above it is listed as being cos(x)
It can be written as:
\frac{d}{dx}sin(x) = cos(x)
Or:
sin(x)’ = cos(x)
Power Rule
Example: What is \frac{d}{dx}x^{3} ?
The question is asking "what is the derivative of x^{3} ?"
We can use the Power Rule, where n=3:
\frac{d}{dx}x^{n} = nx^{n−1}
\frac{d}{dx}x^{3} = 3x^{3−1} = 3x^{2}
(In other words the derivative of x^{3} is 3x^{2})
So it is simply this:
"multiply by power
then reduce power by 1"
It can also be used in cases like this:
Example: What is \frac{d}{dx}(1/x) ?
1/x is also x^{-1}
We can use the Power Rule, where n = −1:
\frac{d}{dx}x^{n} = nx^{n−1}
\frac{d}{dx}x^{-1} = −1x^{-1−1}
= −x^{-2}
= \frac{−1}{x^{2}}
So we just did this:
which simplifies to −1/x^{2}
Multiplication by constant
Example: What is \frac{d}{dx}5x^{3 }?
the derivative of cf = cf’
the derivative of 5f = 5f’
We know (from the Power Rule):
\frac{d}{dx}x^{3} = 3x^{3−1} = 3x^{2}
So:
\frac{d}{dx}5x^{3} = 5\frac{d}{dx}x^{3} = 5 × 3x^{2} = 15x^{2}
Sum Rule
Example: What is the derivative of x^{2}+x^{3 }?
The Sum Rule says:
the derivative of f + g = f’ + g’
So we can work out each derivative separately and then add them.
Using the Power Rule:
- \frac{d}{dx}x^{2} = 2x
- \frac{d}{dx}x^{3} = 3x^{2}
And so:
the derivative of x^{2} + x^{3} = 2x + 3x^{2}
Difference Rule
What we differentiate with respect to doesn't have to be x, it could be anything. In this case v:
Example: What is \frac{d}{dv}(v^{3}−v^{4}) ?
The Difference Rule says
the derivative of f − g = f’ − g’
So we can work out each derivative separately and then subtract them.
Using the Power Rule:
- \frac{d}{dv}v^{3} = 3v^{2}
- \frac{d}{dv}v^{4} = 4v^{3}
And so:
the derivative of v^{3} − v^{4} = 3v^{2} − 4v^{3}
Sum, Difference, Constant Multiplication And Power Rules
Example: What is \frac{d}{dz}(5z^{2} + z^{3} − 7z^{4}) ?
Using the Power Rule:
- \frac{d}{dz}z^{2} = 2z
- \frac{d}{dz}z^{3} = 3z^{2}
- \frac{d}{dz}z^{4} = 4z^{3}
And so:
\frac{d}{dz}(5z^{2} + z^{3} − 7z^{4}) = 5 × 2z + 3z^{2} − 7 × 4z^{3}
= 10z + 3z^{2} − 28z^{3}
Product Rule
Example: What is the derivative of cos(x)sin(x) ?
The Product Rule says:
the derivative of fg = f g’ + f’ g
In our case:
- f = cos
- g = sin
We know (from the table above):
- \frac{d}{dx}cos(x) = −sin(x)
- \frac{d}{dx}sin(x) = cos(x)
So:
the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x)
= cos^{2}(x) − sin^{2}(x)
Quotient Rule
To help you remember:
(\frac{f}{g})’ = \frac{gf’ − fg’}{g^{2}}
The derivative of "High over Low" is:
"Low dHigh minus High dLow, over the line and square the Low"
Example: What is the derivative of cos(x)/x ?
In our case:
- f = cos
- g = x
We know (from the table above):
- f' = −sin(x)
- g' = 1
So:
the derivative of \frac{cos(x)}{x} = \frac{Low dHigh minus High dLow}{square the Low}
= \frac{x(−sin(x)) − cos(x)(1)}{x^{2}}
= −\frac{xsin(x) + cos(x)}{x^{2}}
Reciprocal Rule
Example: What is \frac{d}{dx}(1/x) ?
The Reciprocal Rule says:
the derivative of \frac{1}{f} = \frac{−f’}{f^{2}}
With f(x)= x, we know that f’(x) = 1
So:
the derivative of \frac{1}{x} = \frac{−1}{x^{2}}
Which is the same result we got above using the Power Rule.
Chain Rule
Example: What is \frac{d}{dx}sin(x^{2}) ?
sin(x^{2}) is made up of sin() and x^{2}:
- f(g) = sin(g)
- g(x) = x^{2}
The Chain Rule says:
the derivative of f(g(x)) = f'(g(x))g'(x)
The individual derivatives are:
- f'(g) = cos(g)
- g'(x) = 2x
So:
\frac{d}{dx}sin(x^{2}) = cos(g(x)) (2x)
= 2x cos(x^{2})
Another way of writing the Chain Rule is: \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}
Let's do the previous example again using that formula:
Example: What is \frac{d}{dx}sin(x^{2}) ?
\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}
Let u = x^{2}, so y = sin(u):
\frac{d}{dx} sin(x^{2}) = \frac{d}{du}sin(u)\frac{d}{dx}x^{2}
Differentiate each:
\frac{d}{dx} sin(x^{2}) = cos(u) (2x)
Substitute back u = x^{2} and simplify:
\frac{d}{dx} sin(x^{2}) = 2x cos(x^{2})
Same result as before (thank goodness!)
Another couple of examples of the Chain Rule:
Example: What is \frac{d}{dx}(1/cos(x)) ?
1/cos(x) is made up of 1/g and cos():
- f(g) = 1/g
- g(x) = cos(x)
The Chain Rule says:
the derivative of f(g(x)) = f’(g(x))g’(x)
The individual derivatives are:
- f'(g) = −1/(g^{2})
- g'(x) = −sin(x)
So:
(1/cos(x))’ = \frac{−1}{g(x)^{2}}(−sin(x))
= \frac{sin(x)}{cos^{2}(x)}
Note: \frac{sin(x)}{cos^{2}(x)} is also \frac{tan(x)}{cos(x)} or many other forms.
Example: What is \frac{d}{dx}(5x−2)^{3} ?
The Chain Rule says:
the derivative of f(g(x)) = f’(g(x))g’(x)
(5x−2)^{3} is made up of g^{3} and 5x−2:
- f(g) = g^{3}
- g(x) = 5x−2
The individual derivatives are:
- f'(g) = 3g^{2} (by the Power Rule)
- g'(x) = 5
So:
\frac{d}{dx}(5x−2)^{3} = (3g(x)^{2})(5) = 15(5x−2)^{2}